Context-Free Grammars

Transcription

1 Context-Free Grmmrs

2 Descriing Lnguges We've seen two models for the regulr lnguges: Finite utomt ccept precisely the strings in the lnguge. Regulr expressions descrie precisely the strings in the lnguge. Finite utomt recognize strings in the lnguge. Perform computtion to determine whether specific string is in the lnguge. Regulr expressions mtch strings in the lnguge. Descrie the generl shpe of ll strings in the lnguge.

3 Context-Free Grmmrs A context-free grmmr (or CFG) is n entirely different formlism for defining clss of lnguges. Gol: Give procedure for listing off ll strings in the lnguge. CFGs re est explined y exmple...

4 Arithmetic Expressions Suppose we wnt to descrie ll legl rithmetic expressions using ddition, sutrction, multipliction, nd division. Here is one possile CFG: E int E E Op E E (E) Op + Op - Op * Op / E E Op E E Op (E) E Op (E Op E) E * (E Op E) int * (E Op E) int * (int Op E) int * (int Op int) int * (int + int)

5 Arithmetic Expressions Suppose we wnt to descrie ll legl rithmetic expressions using ddition, sutrction, multipliction, nd division. Here is one possile CFG: E int E E Op E E (E) Op + Op - Op * Op / E E Op E E Op int int Op int int / int

6 Context-Free Grmmrs Formlly, context-free grmmr is collection of four ojects: A set of nonterminl symols (lso clled vriles), A set of terminl symols (the lphet of the CFG) A set of production rules sying how ech nonterminl cn e replced y string of terminls nd nonterminls, nd A strt symol (which must e nonterminl) tht egins the derivtion. E int E E Op E E (E) Op + Op - Op * Op /

7 Some CFG Nottion Cpitl letters in Bold Red Uppercse will represent nonterminls. i.e. A, B, C, D Lowercse letters in lue monospce will represent terminls. i.e. t, u, v, w Lowercse Greek letters in gry itlics will represent ritrry strings of terminls nd nonterminls. i.e. α, γ, ω

8 A Nottionl Shorthnd E int E E Op E E (E) Op + Op - Op * Op /

9 A Nottionl Shorthnd E int E Op E (E) Op + - * /

10 Derivtions E E Op E int (E) Op + * - / E E Op E E Op (E) E Op (E Op E) E * (E Op E) int * (E Op E) int * (int Op E) int * (int Op int) int * (int + int) A sequence of steps where nonterminls re replced y the right-hnd side of production is clled derivtion. If string α derives string ω, we write α * ω. In the exmple on the left, we see E * int * (int + int).

11 The Lnguge of Grmmr If G is CFG with lphet Σ nd strt symol S, then the lnguge of G is the set L(G) = { ω Σ* S * ω } Tht is, L( G) is the set of strings derivle from the strt symol. Note: ω must e in Σ*, the set of strings mde from terminls. Strings involving nonterminls ren't in the lnguge.

12 Context-Free Lnguges A lnguge L is clled context-free lnguge (or CFL) if there is CFG G such tht L = L( G). Questions: Wht lnguges re context-free? How re context-free nd regulr lnguges relted?

13 From Regexes to CFGs CFGs consist purely of production rules of the form A ω. They do not hve the regulr expression opertors * or. However, we cn convert regulr expressions to CFGs s follows: S *

14 From Regexes to CFGs CFGs consist purely of production rules of the form A ω. They do not hve the regulr expression opertors * or. However, we cn convert regulr expressions to CFGs s follows: S A

15 From Regexes to CFGs CFGs consist purely of production rules of the form A ω. They do not hve the regulr expression opertors * or. However, we cn convert regulr expressions to CFGs s follows: S A A A ε

16 From Regexes to CFGs CFGs consist purely of production rules of the form A ω. They do not hve the regulr expression opertors * or. However, we cn convert regulr expressions to CFGs s follows: S ( c*)

17 From Regexes to CFGs CFGs consist purely of production rules of the form A ω. They do not hve the regulr expression opertors * or. However, we cn convert regulr expressions to CFGs s follows: S X X ( c*)

18 From Regexes to CFGs CFGs consist purely of production rules of the form A ω. They do not hve the regulr expression opertors * or. However, we cn convert regulr expressions to CFGs s follows: S X X c*

19 From Regexes to CFGs CFGs consist purely of production rules of the form A ω. They do not hve the regulr expression opertors * or. However, we cn convert regulr expressions to CFGs s follows: S X X c*

20 From Regexes to CFGs CFGs consist purely of production rules of the form A ω. They do not hve the regulr expression opertors * or. However, we cn convert regulr expressions to CFGs s follows: S X X C

21 From Regexes to CFGs CFGs consist purely of production rules of the form A ω. They do not hve the regulr expression opertors * or. However, we cn convert regulr expressions to CFGs s follows: S X X C C Cc ε

22 Regulr Lnguges nd CFLs Theorem: Every regulr lnguge is context-free. Proof Ide: Use the construction from the previous slides to convert regulr expression for L into CFG for L. Prolem Set Exercise: Insted, show how to convert DFA/NFA into CFG.

23 The Lnguge of Grmmr Consider the following CFG G: S S ε Wht strings cn this generte?

24 The Lnguge of Grmmr Consider the following CFG G: S S ε Wht strings cn this generte? S

25 The Lnguge of Grmmr Consider the following CFG G: S S ε Wht strings cn this generte? S

26 The Lnguge of Grmmr Consider the following CFG G: S S ε Wht strings cn this generte? S

27 The Lnguge of Grmmr Consider the following CFG G: S S ε Wht strings cn this generte? S

28 The Lnguge of Grmmr Consider the following CFG G: S S ε Wht strings cn this generte? S

29 The Lnguge of Grmmr Consider the following CFG G: S S ε Wht strings cn this generte? S

30 The Lnguge of Grmmr Consider the following CFG G: S S ε Wht strings cn this generte? S

31 The Lnguge of Grmmr Consider the following CFG G: S S ε Wht strings cn this generte? S

32 The Lnguge of Grmmr Consider the following CFG G: S S ε Wht strings cn this generte?

33 The Lnguge of Grmmr Consider the following CFG G: S S ε Wht strings cn this generte?

34 The Lnguge of Grmmr Consider the following CFG G: S S ε Wht strings cn this generte? L(G) = { n n n N }

35 All Lnguges Regulr Lnguges CFLs

36 Why the Extr Power? Why do CFGs hve more power thn regulr expressions? Intuition: Derivtions of strings hve unounded memory. S S ε

37 Why the Extr Power? Why do CFGs hve more power thn regulr expressions? Intuition: Derivtions of strings hve unounded memory. S S ε

38 Why the Extr Power? Why do CFGs hve more power thn regulr expressions? Intuition: Derivtions of strings hve unounded memory. S S ε S

39 Why the Extr Power? Why do CFGs hve more power thn regulr expressions? Intuition: Derivtions of strings hve unounded memory. S S ε S

40 Why the Extr Power? Why do CFGs hve more power thn regulr expressions? Intuition: Derivtions of strings hve unounded memory. S S ε S

41 Why the Extr Power? Why do CFGs hve more power thn regulr expressions? Intuition: Derivtions of strings hve unounded memory. S S ε S

42 Why the Extr Power? Why do CFGs hve more power thn regulr expressions? Intuition: Derivtions of strings hve unounded memory. S S ε S

43 Why the Extr Power? Why do CFGs hve more power thn regulr expressions? Intuition: Derivtions of strings hve unounded memory. S S ε S

44 Why the Extr Power? Why do CFGs hve more power thn regulr expressions? Intuition: Derivtions of strings hve unounded memory. S S ε S

45 Why the Extr Power? Why do CFGs hve more power thn regulr expressions? Intuition: Derivtions of strings hve unounded memory. S S ε S

46 Why the Extr Power? Why do CFGs hve more power thn regulr expressions? Intuition: Derivtions of strings hve unounded memory. S S ε

47 Why the Extr Power? Why do CFGs hve more power thn regulr expressions? Intuition: Derivtions of strings hve unounded memory. S S ε

48 Time-Out for Announcements!

49 Prolem Sets Prolem Set Six ws due t the strt of clss; feel free to turn it in y the strt of Mondy's lecture using lte dys. Prolem Set Seven goes out tody. It's due on Fridy of next week. Ply round with the limits of regulr lnguges! Explore context-free grmmrs! See the interply of utomt nd CFGs! As lwys, feel free to stop y office hours or sk questions on Pizz.

50 A Reminder: The Honor Code This is the point in the qurter where we strt to see lot of cheting cses. Plese tke the following to hert: It is not the end of the world if you cn't figure out ll the prolems on prolem set. Ech prolem set is worth something like 3% of your totl grde in this course. Skipping prolem set is not going to tnk your grde. If you know someone in this clss who is unhelthily stressed out, plese rech out to them. We ll need to look out for ech other nd tke cre of ech other. If you know someone who is relly hurting, e there for them. If you re feeling overwhelmed y this clss, plese come tlk to us. This clss is hrd, ut it's not supposed to mke you suffer. If things ren't going well, we'd e hppy to discuss your options. You re not your grdes. You re humn eing. I mentioned this erlier, ut it is not the end of the world if you get low grde, withdrw from this clss, or fil this clss. It does not reflect poorly on you. It does not mke you worthless. From experience, osessing over your grdes will mke you miserle in the long term. Plese don't mke the sme mistkes I mde.

51 Your Questions!

52 I m feeling overwhelmed right now with ll my clsses. Second wve of midterms hve lredy strted for me, nd I m finding it hrd to lnce studying for the midterms nd stying on top of the dy to dy mteril for ll my clsses. Any tips? I'm sorry to her tht. Tht cn e relly rough. There is I'm sorry to her tht. Tht cn e relly rough. There is no one size fits ll solution here. Your gol is to mke no one size fits ll solution here. Your gol is to mke sure tht not ll of the following re true: sure tht not ll of the following re true: you hve too mny things to do, you hve too mny things to do, you hve to do ll of them well, you hve to do ll of them well, you don't enjoy them, you don't enjoy them, you hve limited time to do them, nd you hve limited time to do them, nd this process repets. this process repets. Try ddressing ech of these independently. If you cn Try ddressing ech of these independently. If you cn ddress the root cuses of ech, you will proly end up ddress the root cuses of ech, you will proly end up lot hppier. lot hppier.

53 Three Questions Wht is something tht you now know tht, t the strt of the qurter, you knew you didn't know? Wht is something tht you now know tht, t the strt of the qurter, you didn't know you didn't know? Wht is something tht you don't know tht, t the strt of the qurter, you didn't know you didn't know?

54 I herd rumors tht there re plns to mke the CS deprtment more difficult to get into, ecuse there's so mny people trying to mjor in CS. Is this true? Also, wht re some of the plns out there to tckle the influx of so mny undergrds? I hven't herd nything like this it goes ginst so much of I hven't herd nything like this it goes ginst so much of wht mkes Stnford Stnford. We've hd lot of discussions wht mkes Stnford Stnford. We've hd lot of discussions out how to del with logisticl issues from lrge clss sizes nd out how to del with logisticl issues from lrge clss sizes nd re considering things like incresing the numer of offerings of re considering things like incresing the numer of offerings of ech clss, hving multiple people tech the sme clss ech ech clss, hving multiple people tech the sme clss ech qurter, hiring more stff, etc., ut nothing like this. qurter, hiring more stff, etc., ut nothing like this. We'd consider it huge filure on our prt if we mde it hrder We'd consider it huge filure on our prt if we mde it hrder to get into CS. Tht would undo so much of the work we've done to get into CS. Tht would undo so much of the work we've done in mking CS more ccessile nd interesting nd would totlly in mking CS more ccessile nd interesting nd would totlly poison the culture in the deprtment. poison the culture in the deprtment.

55 Bck to CS103!

56 Designing CFGs Like designing DFAs, NFAs, nd regulr expressions, designing CFGs is crft. When thinking out CFGs: Think recursively: Build up igger structures from smller ones. Hve construction pln: Know in wht order you will uild up the string. Store informtion in nonterminls: Hve ech nonterminl correspond to some useful piece of informtion.

57 Designing CFGs Let Σ = {, } nd let L = {w Σ* w is plindrome } We cn design CFG for L y thinking inductively: Bse cse: ε,, nd re plindromes. If ω is plindrome, then ω nd ω re plindromes. S ε S S

58 Designing CFGs Let Σ = {(, )} nd let L = {w Σ* w is string of lnced prentheses } Some smple strings in L: ((())) (())() (()())(()()) ((((()))(()))) ε ()()

59 Designing CFGs Let Σ = {(, )} nd let L = {w Σ* w is string of lnced prentheses } Let's think out this recursively. Bse cse: the empty string is string of lnced prentheses. Recursive step: Look t the closing prenthesis tht mtches the first open prenthesis. ((()(()))(()))(())((()))

60 Designing CFGs Let Σ = {(, )} nd let L = {w Σ* w is string of lnced prentheses } Let's think out this recursively. Bse cse: the empty string is string of lnced prentheses. Recursive step: Look t the closing prenthesis tht mtches the first open prenthesis. (( ( ) ( ( ) ) )( ( ) ))(())(( ( ) ))

61 Designing CFGs Let Σ = {(, )} nd let L = {w Σ* w is string of lnced prentheses } Let's think out this recursively. Bse cse: the empty string is string of lnced prentheses. Recursive step: Look t the closing prenthesis tht mtches the first open prenthesis. (( ( ) ( ( ) ) )( ( ) ))(())(( ( ) ))

62 Designing CFGs Let Σ = {(, )} nd let L = {w Σ* w is string of lnced prentheses } Let's think out this recursively. Bse cse: the empty string is string of lnced prentheses. Recursive step: Look t the closing prenthesis tht mtches the first open prenthesis. ( ( ) ( ( ) ) )( ( ) ) (())(( ( ) ))

63 Designing CFGs Let Σ = {(, )} nd let L = {w Σ* w is string of lnced prentheses } Let's think out this recursively. Bse cse: the empty string is string of lnced prentheses. Recursive step: Look t the closing prenthesis tht mtches the first open prenthesis. Removing the first prenthesis nd the mtching prenthesis forms two new strings of lnced prentheses. S (S)S ε

64 Designing CFGs: A Cvet Let Σ = {, } nd let L = {w Σ* w hs the sme numer of 's nd 's } Is this CFG for L? S S S ε Cn you derive the string?

65 Designing CFGs: A Cvet When designing CFG for lnguge, mke sure tht it genertes ll the strings in the lnguge nd never genertes string outside the lnguge. The first of these cn e tricky mke sure to test your grmmrs! You'll design your own CFG for this lnguge on the next prolem set.

66 CFG Cvets II Is the following grmmr CFG for the lnguge { n n n N }? S S Wht strings cn you derive? Answer: None! Wht is the lnguge of the grmmr? Answer: Ø When designing CFGs, mke sure your recursion ctully termintes!

67 CFG Cvets III When designing CFGs, rememer tht ech nonterminl cn e expnded out independently of the others. Let Σ = {, } nd let L = {n n n N }. Is the following CFG for L? S X X X X ε S X X X X X X X X

68 Finding Build Order Let Σ = {, } nd let L = { n n n N }. To uild CFG for L, we need to e more clever with how we construct the string. If we uild the strings of 's independently of one nother, then we cn't enforce tht they hve the sme length. Ide: Build oth strings of 's t the sme time. Here's one possile grmmr sed on tht ide: S S S S S S

69 Function Prototypes Let Σ = {void, int, doule, nme, (, ),,, ;}. Let's write CFG for C-style function prototypes! Exmples: void nme(int nme, doule nme); int nme(); int nme(doule nme); int nme(int, int nme, int); void nme(void);

70 Function Prototypes Here's one possile grmmr: S Ret nme (Args); Ret Type void Type int doule Args ε void ArgList ArgList OneArg ArgList, OneArg OneArg Type Type nme Fun question to think out: wht chnges would you need to mke to support pointer types?

71 Summry of CFG Design Tips Look for recursive structures where they exist: they cn help guide you towrd solution. Keep the uild order in mind often, you'll uild two totlly different prts of the string concurrently. Usully, those prts re uilt in opposite directions: one's uilt left-to-right, the other right-to-left. Use different nonterminls to represent different structures.

72 Applictions of Context-Free Grmmrs

73 CFGs for Progrmming Lnguges BLOCK STMT { STMTS } STMTS ε STMT STMTS STMT EXPR; if (EXPR) BLOCK while (EXPR) BLOCK do BLOCK while (EXPR); BLOCK EXPR identifier constnt EXPR + EXPR EXPR EXPR EXPR * EXPR...

74 Grmmrs in Compilers One of the key steps in compiler is figuring out wht progrm mens. This is usully done y defining grmmr showing the high-level structure of progrmming lnguge. There re certin clsses of grmmrs (LL(1) grmmrs, LR(1) grmmrs, LALR(1) grmmrs, etc.) for which it's esy to figure out how prticulr string ws derived. Tools like ycc or ison utomticlly generte prsers from these grmmrs. Curious to lern more? Tke CS143!

75 Nturl Lnguge Processing By uilding context-free grmmrs for ctul lnguges nd pplying sttisticl inference, it's possile for computer to recover the likely mening of sentence. In fct, CFGs were first clled phrse-structure grmmrs nd were introduced y Nom Chomsky in his seminl work Syntctic Structures. They were then dpted for use in the context of progrmming lnguges, where they were clled Bckus- Nur forms. Stnford's CoreNLP project is one plce to look for n exmple of this. Wnt to lern more? Tke CS124 or CS224N!

76 Next Time Turing Mchines Wht does computer with unounded memory look like? How do you progrm them?